Charles Werneth

Numerical Gram–Schmidt orthonormalization

A numerical Gram–Schmidt orthonormalization procedure is presented for constructing an orthonormal basis function set from a non-orthonormal set, when the number of basis functions is large. This method will provide a pedagogical illustration of the Gram–Schmidt procedure and can be presented in classes on numerical methods or computational physics.

Elastic differential cross sections for space radiation applications

The eikonal, partial wave (PW) Lippmann-Schwinger, and three-dimensional Lippmann- Schwinger (LS3D) methods are compared for nuclear reactions that are relevant for space radiation applications. Numerical convergence of the eikonal method is readily achieved when exact formulas of the optical potential are used for light nuclei (A

Estimates of Signicant Post-Carrington Solar Particle Event Radiation Exposures on Mars

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Estimates of Carrington-Class Solar Particle Event Radiation Exposures on Mars Behind Polyethylene Shielding

16), and the momentum-space representation of the optical potential is used for heavier nuclei. The PW solution method is known to be numerically unstable for systems that require a large number of partial waves, and, as a result, the LS3D method is employed. The effect of relativistic kinematics is studied with the PW and LS3D methods and is compared to eikonal results. It is recommended that the LS3D method be used for high energy nucleon-nucleus reactions and nucleus-nucleus reactions at all energies because of its rapid numerical convergence and stability.

Variational method using basis functions

1864, 1878, 1894, 1895, and 1896), are made for male and female crew members located at the mean surface elevation and at an altitude of 8 km in the Martian atmosphere. The incident solar particle event proton energy distributions for these events are assumed to be similar to that of the Band function t of the February 1956 event. Radiation exposure estimates were performed using NASA’s On-Line Tool for the Assessment of Radiation in Space (OLTARIS). The HZETRN (High charge (Z) and Energy TRaNsport) space radiation transport code, which is incorporated into OLTARIS, was used to describe the transport of incident protons and any secondary particles generated by their interactions with the atmosphere of Mars, through spacesuit, surface lander, or permanent habitat shielding, and body organ self-shielding. Estimates of eective dose and organ dose are made using the Computerized Anatomical Male (CAM), Computerized Anatomical Female (CAF), Male Adult voXel (MAX), and Female Adult voXel (FAX) human geometry models. The predicted exposures are compared with current NASA Permissible Exposure Limits (PELs).

Elastic Differential Cross Sections

Radiation exposure estimates for crew members on the surface of Mars are made for solar particle event proton radiation environments comparable to the Carrington event of 1859. We assume that the proton energy distribution for this Carrington-type event is similar to that of the Band Function fit of the February 1956 event. In this work we use the BRYNTRN radiation transport code, originally developed at NASA Langley Research Center. The Computerized Anatomical Male and Female human geometry models were utilized to estimate exposures for polyethylene shield areal densities similar to those provided by a spacesuit, a surface lander, and a permanent habitat located at various altitudes in the Mars atmosphere. Comparisons of the predicted organ exposures are made with previously reported values obtained for aluminum shielding and current NASA Permissible Exposure Limits (PELs).

Correlated Uncertainties in Radiation Shielding Effectiveness

The Gaussian, exponential and Laguerre basis functions are examined in a variational calculation of energies and wavefunctions. The Laguerre basis set is already orthonormal and complete, but the Gaussian and exponential basis sets are not orthonormal. We used the linear and Coulomb potentials to test these basis functions. Calculations are performed in both position and momentum space. We also present the results with relativistic kinematics in the momentum space calculation. The Gram–Schmidt procedure is used to orthonormalize the Gaussian and exponential basis sets before using them in the calculations. We show that in the case of a pure linear potential, the orthonormal basis constructed from the Gaussian functions performs much better than the exponential and Laguerre basis for the same number of orthogonal functions. For Coulomb-like potentials, the exponential basis performs better than the other two for the same number of basis functions. The advantage of using these simple basis functions is that for the potentials that we examined, one can approach the lower bound of the low-lying states with very few basis functions.